A145885 a(n) = (n-1)^2*binomial(2n,n)/(2*(n+1)).

0, 1, 10, 63, 336, 1650, 7722, 35035, 155584, 680238, 2939300, 12584726, 53488800, 225990180, 950094810, 3977737875, 16594533120, 69018792150, 286296636780, 1184823735810, 4893253404000, 20171905282620, 83020426503300

Offset: 1

Author

Emeric Deutsch, Nov 06 2008

Keywords

• nonn
• easy

Comments

a(n) = sum of valley abscissae in all Dyck paths of semilength n minus number of valleys in all Dyck paths of semilength (n). Example: a(3)=10; indeed, the Dyck paths of semilength 3, followed by their valley abscissae are UDUDUD (2,4), UDUUDD (2), UUDDUD (4), UUDUDD (3), UUUDDD ( ); therefore a(3)=2+4+2+4+3 - 5 = 10. Instead of Dyck paths one can consider Dyck words; then sum of valley abscissae corresponds to major index and number of valleys to number of descents.

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 d.

Crossrefs

Cf. A000108, A002740, A002054, A145884.

Programs

Maple

seq((1/2)*(n-1)^2*binomial(2*n,n)/(n+1),n=1..24);

Mathematica

Table[CatalanNumber[n]*(n - 1)^2/2, {n, 1, 23}] (* Zerinvary Lajos, Jul 08 2009 *)

Formula

a(n) = A002740(n+1) - A002054(n-1) (n >= 2).

a(n) = Sum_{k=0..(n-1)^2} k*A145884(n,k) for n >= 1.

a(n) = (n-1)^2*Cat(n)/2, where Cat(n)=binomial(2n,n)/(n+1)=A000108(n) are the Catalan numbers.

G.f.: 4*z^2*(8*z - 1 + 3*sqrt(1-4*z))/((1 + sqrt(1-4*z))^3*(1-4*z)^(3/2)).

D-finite with recurrence (n+1)*(n-2)^2*a(n) - 2*(2*n-1)*(n-1)^2*a(n-1) = 0. - R. J. Mathar, Aug 10 2017

E.g.f.: (exp(2*x) * ((1 + 2*x) * BesselI(0,2*x) - 2 * (2 - x) * BesselI(1,2*x)) - 1) / 2. - Ilya Gutkovskiy, Nov 03 2021